Theme Essential Question: How are functions used to describe relationships that happen in our lives?
Essential Questions: How do you write an equation for a function given a table, graph, or description?
Standards
8. EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8. F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8. F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Objectives
The student will use any combination of functions representations (verbal description, table, graph and/or equation) to compare and contrast the data provided.
The student will draw conclusions using a combination of functional representations.
In Grade 8, students focus on linear equations and functions. Nonlinear functions are used for comparison.
Students will need many opportunities and examples to figure out the meaning of y = mx + b. What does m mean? What does b mean? They should be able to “see” m and b in graphs, tables, and formulas or equations, and they need to be able to interpret those values in contexts.
Use graphing calculators and web resources to explore linear and non-linear functions. Provide context as much as possible to build understanding of slope and y-intercept in a graph, especially for those patterns that do not start with an initial value of 0.
From a variety of representations of functions, students should be able to classify and describe the function as linear or non-linear, increasing or decreasing. Provide opportunities for students to share their ideas with other students and create their own examples for their classmates.
Use the slope of the graph and similar triangle arguments to call attention to not just the change in x or y, but also to the rate of change, which is a ratio of the two.
Assessment Product Entry #6
Students will continue creating a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entry provided follows the theme of construction repair and remodel.
Provide students with a function in symbolic form and ask them to create a problem situation in word to match the function. They are to use their construction repair and remodel theme.
Given a graph, have students create a scenario that would fit the graph. They are to use their construction repair and remodel theme.
OR
Cross-Curriculum
Give students opportunities to gather their own data or graphs in contexts they understand. Connect with your team teachers to provide scenarios for students to measure, collect data, graph data, and look for patterns. Have students prepare a lab report with generalize and symbolically represent of their patterns. Have them draw graphs (qualitatively, based upon experience) representing their real-life situations with which they are familiar. Use probing questions to determine which input values make sense in the problem situations.
Key Questions
How do you compare two different proportional relationships represented in different ways?
What properties facilitate the comparison of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)?
Observable Student Behaviors
The student can use any combination of functional representation (algebraically, graphically, numerically in tables, or by verbal descriptions) to compare, contrast or analyze a problem.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary Linear function, slope-intercept form, rate of change, unit rate
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 2-6 p.53-56
ABC Mastering the Common Core in Mathematics Chapter 5-15, p. 66-69
Gizmo Lessons
Direct Variation
Adjust the constant of variation and explore how the graph of the direct variation function changes in response.
Estimating Population Size
Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake.
Distance-Time Graph
Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. (Student Exploration Sheet & Teacher Guide Available)
Modeling Linear Systems-Activity A
Experiment with a system of two lines representing a cat‑and‑mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real‑world meaning to slope, y‑intercept, and the intersection of lines.
Simple and Compound Interest
Find the current balance and the interest charged on an investment using the graph of the interest function and by directly calculating.
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
8.EE.5 p. 189
8.F.2 p. 185
8.F.4 Activity 1p. 191, Activity 2 p. 192
JBHM PreAlgebra, GP 2, p329-350
Glencoe Pre Algebra Chapter 8, Lesson 7, p. 404-408
Glencoe Algebra I Chapter 5, Lesson 4 , p.280-285
Extension, Point-Slope form, p.286-291
Extension, Parallel & Perpendicular, p. 292-297
Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
Grade: 8 Unit: 3 Week: 6 Dates: 1/14-1/18
Content: Comparing Functions
Theme Essential Question:
How are functions used to describe relationships that happen in our lives?
Essential Questions:
How do you write an equation for a function given a table, graph, or description?
Standards
Objectives
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials.
**http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf**
(Taken from **Ohio Dept of Education Mathematics Model Curriculum 6-28-2011** )
In Grade 8, students focus on linear equations and functions. Nonlinear functions are used for comparison.
Students will need many opportunities and examples to figure out the meaning of y = mx + b. What does m mean? What does b mean? They should be able to “see” m and b in graphs, tables, and formulas or equations, and they need to be able to interpret those values in contexts.
Use graphing calculators and web resources to explore linear and non-linear functions. Provide context as much as possible to build understanding of slope and y-intercept in a graph, especially for those patterns that do not start with an initial value of 0.
From a variety of representations of functions, students should be able to classify and describe the function as linear or non-linear, increasing or decreasing. Provide opportunities for students to share their ideas with other students and create their own examples for their classmates.
Use the slope of the graph and similar triangle arguments to call attention to not just the change in x or y, but also to the rate of change, which is a ratio of the two.
Assessment
Product
Entry #6
- Students will continue creating a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entry provided follows the theme of construction repair and remodel.
- Provide students with a function in symbolic form and ask them to create a problem situation in word to match the function. They are to use their construction repair and remodel theme.
- Given a graph, have students create a scenario that would fit the graph. They are to use their construction repair and remodel theme.
ORCross-Curriculum
Key Questions
Observable Student Behaviors
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
Linear function, slope-intercept form, rate of change, unit rate
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 2-6 p.53-56
- ABC Mastering the Common Core in Mathematics Chapter 5-15, p. 66-69
- Gizmo Lessons
- Direct Variation
Adjust the constant of variation and explore how the graph of the direct variation function changes in response.- Estimating Population Size
Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake.- Distance-Time Graph
Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. (Student Exploration Sheet & Teacher Guide Available)- Modeling Linear Systems-Activity A
Experiment with a system of two lines representing a cat‑and‑mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real‑world meaning to slope, y‑intercept, and the intersection of lines.- Simple and Compound Interest
Find the current balance and the interest charged on an investment using the graph of the interest function and by directly calculating.Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
Homework
http://www.kutasoftware.com/free.html to print assignments on variety of topics
See appropriate Glencoe, OnCore, JBHM, and ABC materials
Terminology for Teachers
Ethnicity/Culture | Immigration/Migration | Intercultural Competence | Socialization | Racism/Discrimination
High Yield Strategies
Similarities/Differences | Summarizing/Notetaking | Reinforcing/Recognition | Homework/Practice |
Non-Linguistic representation | Cooperative Learning | Objectives/Feedback |
Generating-Testing Hypothesis | Cues, Questions, Organizers
Resources
Professional Texts
Literary Texts
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Art, Music, and Media
Manipulatives
Games
Videos
SMART Board Lessons, Promethean Lessons
Websites
Other Activities, etc.
Language
Arts
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Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Matrix
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Home K-2
Home 3-6
Home 6-8
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6